Analytical Properties of k-Generalized Digamma and Polygamma Functions Derived From Dilcher-Type Gamma Functions

This study presents a detailed investigation into the analytical properties of the generalized Gamma function introduced by Dilcher. We establish a fundamental recurrence relation and derive novel reflection formula for this generalized function, extending the classical identities known for the Euler Gamma function. Further, we analyze the generalized digamma function, the logarithmic derivative of the generalized Gamma function, and obtain a significant difference equation that characterizes its behavior. Explicit expressions for the evaluation of this generalized digamma function at half-integer arguments are provided. The study also introduces the natural extension to generalized polygamma functions and reports on several special values for both classical and generalized cases. To complement the theoretical analysis, a graphical representation of the generalized digamma function is included, alongside supporting numerical computations that illustrate its properties and validate the derived results. Our findings provide a comprehensive framework that enhances the understanding of these generalized special functions and underscores their potential for broader application in mathematical physics and number theory.